As I understand, the Ethereum project provides an innovative technology for allowing developers to build distributed apps that can insert smart contracts in their blockchain and execute these on their distributed network.

Why do they need a coin for any of this? Couldn't they just have an auction market place where developers bid to execute apps and miners accept bids?

Having a cost associated with taking actions places a necessary limit on the size of the blockchain.

Number Two:

It allows for the exchange of value in unique ways. For example, requiring m of n parties to consent for funds to be exchanged. Or even using time-based encryption to release funds after time t.

Say you want to leave money for your granddaughter Suzie, but you don't want your evil son-in-law Stewie to keep it for himself.

Normally you'd have to have a bank or government to hold your money for some period of time, while they line their pockets with interest made on your savings. By writing a contract that requires some percent of the family to withdraw funds after 30 years, you can rest easy knowing little Suzie will get the money you left for her.

That's a contrived example, but you get the idea. Think of a multisignature wallet on steroids. That's an Ethereum smart contract that, by definition requires a store of value.

Number Three:

Ether can act as a reserve currency for ICOs. Random companies with nothing but a whitepaper and pretty websites are launching ICOs with Ethereum tokens, and the majority are worthless.

The list goes on. Decentralized applications just play really nicely with a decentralized currency.

Couldn't they just have an auction market place where developers bid to execute apps and miners accept bids?

No.

Why would you trust the developers to decide who runs the apps? The whole point of the system is that all you have to trust is that the code does what you want and that the Ethereum system as a whole works.

Suppose there's some contract that both you and I want to execute. And say the results are different if you execute the contract first than if I execute the contract first. How do we ensure everyone agrees on who executed the contract first?